The integers relatively prime to m form a group under multiplication modulo m, called the prime residue group. This chapter describes the functions that deal with this group.
The first section describes the function that computes the set of representatives of the group (see PrimeResidues).
The next sections describe the functions that compute the size and the exponent of the group (see Phi and Lambda).
The next section describes the function that computes the order of an element in the group (see OrderMod).
The next section describes the functions that test whether a residue generates the group or computes a generator of the group, provided it is cyclic (see IsPrimitiveRootMod, PrimitiveRootMod).
The next section describes the functions that test whether an element is a square in the group (see Jacobi and Legendre).
The next sections describe the functions that compute general roots in the group (see RootMod and RootsUnityMod).
All these functions are in the file LIBNAME/"numtheor.g"
.
PrimeResidues( m )
PrimeResidues
returns the set of integers from the range 0..Abs(m)-1
that are relatively prime to the integer m.
Abs(m) must be less than 228, otherwise the set would probably be too large anyhow.
The integers relatively prime to m form a group under multiplication modulo m, called the prime residue group. φ(m) (see Phi) is the order of this group, λ(m) (see Lambda) the exponent. If and only if m is 2, 4, an odd prime power pe, or twice an odd prime power 2 pe, this group is cyclic. In this case the generators of the group, i.e., elements of order φ(m), are called primitive roots (see IsPrimitiveRootMod, PrimitiveRootMod).
gap> PrimeResidues( 0 ); [ ] gap> PrimeResidues( 1 ); [ 0 ] gap> PrimeResidues( 20 ); [ 1, 3, 7, 9, 11, 13, 17, 19 ]
Phi( m )
Phi
returns the value of the Euler totient function φ(m) for the
integer m. φ(m) is defined as the number of positive integers
less than or equal to m that are relatively prime to m.
Suppose that m = p1e1 p2e2 ... pkek. Then φ(m) is p1e1-1 (p1-1) p2e2-1 (p2-1) ... pkek-1 (pk-1). It follows that m is a prime if and only if φ(m) = m - 1.
The integers relatively prime to m form a group under multiplication
modulo m, called the prime residue group. It can be computed with
PrimeResidues
(see PrimeResidues). φ(m) is the order of this
group, λ(m) (see Lambda) the exponent. If and only if m is
2, 4, an odd prime power pe, or twice an odd prime power 2 pe, this
group is cyclic. In this case the generators of the group, i.e.,
elements of order φ(m), are called primitive roots (see
IsPrimitiveRootMod, PrimitiveRootMod).
Phi
usually spends most of its time factoring m (see FactorsInt).
gap> Phi( 12 ); 4 gap> Phi( 2^13-1 ); 8190 # which proves that 213-1 is a prime gap> Phi( 2^15-1 ); 27000
Lambda( m )
Lambda
returns the exponent of the group of relatively prime residues
modulo the integer m.
λ(m) is the smallest positive integer l such that for every a relatively prime to m we have al=1 mod m. Fermat's theorem asserts aφ(m)=1 mod m, thus λ(m) divides φ(m) (see Phi).
Carmichael's theorem states that λ can be computed as follows λ(2)=1, λ(4)=2 and λ(2e) = 2e-2 if 3 <= e, λ(pe) = (p-1) pe-1 (= φ(pe)) if p is an odd prime, and λ(n m) = Lcm(λ(n),λ(m)) if n, m are relatively prime.
Composites for which λ(m) divides m - 1 are called Carmichaels. If 6k+1, 12k+1 and 18k+1 are primes their product is such a number. It is believed but unproven that there are infinitely many Carmichaels. There are only 1547 Carmichaels below 1010 but 455052511 primes.
The integers relatively prime to m form a group under multiplication
modulo m, called the prime residue group. It can be computed with
PrimeResidues
(see PrimeResidues). φ(m) (see Phi) is the
order of this group, λ(m) the exponent. If and only if m is 2,
4, an odd prime power pe, or twice an odd prime power 2 pe, this
group is cyclic. In this case the generators of the group, i.e.,
elements of order φ(m), are called primitive roots (see
IsPrimitiveRootMod, PrimitiveRootMod).
Lambda
usually spends most of its time factoring m (see
FactorsInt).
gap> Lambda( 10 ); 4 gap> Lambda( 30 ); 4 gap> Lambda( 561 ); 80 # 561 is the smallest Carmichael number
OrderMod( n, m )
OrderMod
returns the multiplicative order of the integer n modulo the
positive integer m. If n is less than 0 or larger than m it is
replaced by its remainder. If n and m are not relatively prime the
order of n is not defined and OrderMod
will return 0.
If n and m are relatively prime the multiplicative order of n modulo m is the smallest positive integer i such that ni = 1 mod m. Elements of maximal order are called primitive roots (see Phi).
OrderMod
usually spends most of its time factoring m and φ(m)
(see FactorsInt).
gap> OrderMod( 2, 7 ); 3 gap> OrderMod( 3, 7 ); 6 # 3 is a primitive root modulo 7
IsPrimitiveRootMod( r, m )
IsPrimitiveRootMod
returns true
if the integer r is a primitive
root modulo the positive integer m and false
otherwise. If r is
less than 0 or larger than m it is replaced by its remainder.
The integers relatively prime to m form a group under multiplication
modulo m, called the prime residue group. It can be computed with
PrimeResidues
(see PrimeResidues). φ(m) (see Phi) is the
order of this group, λ(m) (see Lambda) the exponent. If and
only if m is 2, 4, an odd prime power pe, or twice an odd prime
power 2 pe, this group is cyclic. In this case the generators of the
group, i.e., elements of order φ(m), are called primitive roots
(see also PrimitiveRootMod).
gap> IsPrimitiveRootMod( 2, 541 ); true gap> IsPrimitiveRootMod( -539, 541 ); true # same computation as above gap> IsPrimitiveRootMod( 4, 541 ); false gap> ForAny( [1..29], r -> IsPrimitiveRootMod( r, 30 ) ); false # there does not exist a primitive root modulo 30
PrimitiveRootMod( m )
PrimitiveRootMod( m, start )
PrimitiveRootMod
returns the smallest primitive root modulo the
positive integer m and false
if no such primitive root exists. If
the optional second integer argument start is given PrimitiveRootMod
returns the smallest primitive root that is strictly larger than start.
The integers relatively prime to m form a group under multiplication
modulo m, called the prime residue group. It can be computed with
PrimeResidues
(see PrimeResidues). φ(m) (see Phi) is the
order of this group, λ(m) (see Lambda) the exponent. If and
only if m is 2, 4, an odd prime power pe, or twice an odd prime
power 2 pe, this group is cyclic. In this case the generators of the
group, i.e., elements of order φ(m), are called primitive roots
(see also IsPrimitiveRootMod).
gap> PrimitiveRootMod( 409 ); 21 # largest primitive root for a prime less than 2000 gap> PrimitiveRootMod( 541, 2 ); 10 gap> PrimitiveRootMod( 337, 327 ); false # 327 is the largest primitive root mod 337 gap> PrimitiveRootMod( 30 ); false # the exists no primitive root modulo 30
Jacobi( n, m )
Jacobi
returns the value of the Jacobi symbol of the integer n
modulo the integer m.
Suppose that m = p1 p2 .. pk as a product of primes, not necessarily distinct. Then for n relatively prime to m the Jacobi symbol is defined by J(n/m) = L(n/p1) L(n/p2) .. L(n/pk), where L(n/p) is the Legendre symbol (see Legendre). By convention J(n/1) = 1. If the gcd of n and m is larger than 1 we define J(n/m) = 0.
If n is an quadratic residue modulo m, i.e., if there exists an r such that r2 = n mod m then J(n/m) = 1. However J(n/m) = 1 implies the existence of such an r only if m is a prime.
Jacobi
is very efficient, even for large values of n and m, it is
about as fast as the Euclidean algorithm (see Gcd).
gap> Jacobi( 11, 35 ); 1 # 92 = 11 mod 35 gap> Jacobi( 6, 35 ); -1 # thus there is no r such that r2 = 6 mod 35 gap> Jacobi( 3, 35 ); 1 # even though there is no r with r2 = 3 mod 35
Legendre( n, m )
Legendre
returns the value of the Legendre symbol of the integer n
modulo the positive integer m.
The value of the Legendre symbol L(n/m) is 1 if n is a quadratic residue modulo m, i.e., if there exists an integer r such that r2 = n mod m and -1 otherwise.
If a root of n exists it can be found by RootMod
(see RootMod).
While the value of the Legendre symbol usually is only defined for m a prime, we have extended the definition to include composite moduli too. The Jacobi symbol (see Jacobi) is another generalization of the Legendre symbol for composite moduli that is much cheaper to compute, because it does not need the factorization of m (see FactorsInt).
gap> Legendre( 5, 11 ); 1 # 42 = 5 mod 11 gap> Legendre( 6, 11 ); -1 # thus there is no r such that r2 = 6 mod 11 gap> Legendre( 3, 35 ); -1 # thus there is no r such that r2 = 3 mod 35
RootMod( n, m )
RootMod( n, k, m )
In the first form RootMod
computes a square root of the integer n
modulo the positive integer m, i.e., an integer r such that r2 = n
mod m. If no such root exists RootMod
returns false
.
A root of n exists only if Legendre(n,m) = 1
(see Legendre).
If m has k different prime factors then there are 2k different
roots of n mod m. It is unspecified which one RootMod
returns.
You can, however, use RootsUnityMod
(see RootsUnityMod) to compute
the full set of roots.
In the second form RootMod
computes a kth root of the integer n
modulo the positive integer m, i.e., an integer r such that rk = n
mod m. If no such root exists RootMod
returns false
.
In the current implementation k must be a prime.
RootMod
is efficient even for large values of m, actually most time
is usually spent factoring m (see FactorsInt).
gap> RootMod( 64, 1009 );
1001 # note RootMod
does not return 8 in this case but -8
gap> RootMod( 64, 3, 1009 );
518
gap> RootMod( 64, 5, 1009 );
656
gap> List( RootMod( 64, 1009 ) * RootsUnityMod( 1009 ),
> x -> x mod 1009 );
[ 1001, 8 ] # set of all square roots of 64 mod 1009
RootsUnityMod( m )
RootsUnityMod( k, m )
In the first form RootsUnityMod
computes the square roots of 1 modulo
the integer m, i.e., the set of all positive integers r less than
n such that r2 = 1 mod m.
In the second form RootsUnityMod
computes the kth roots of 1 modulo
the integer m, i.e., the set of all positive integers r less than
n such that rk = 1 mod m.
In general there are kn such roots if the modulus m has n different prime factors p such that p = 1 mod k. If k2 divides m then there are kn+1 such roots; and especially if k = 2 and 8 divides m there are 2n+2 such roots.
If you are interested in the full set of roots of another number instead
of 1 use RootsUnityMod
together with RootMod
(see RootMod).
In the current implementation k must be a prime.
RootsUnityMod
is efficient even for large values of m, actually most
time is usually spent factoring m (see FactorsInt).
gap> RootsUnityMod(7*31); [ 1, 92, 125, 216 ] gap> RootsUnityMod(3,7*31); [ 1, 25, 32, 36, 67, 149, 156, 191, 211 ] gap> RootsUnityMod(5,7*31); [ 1, 8, 64, 78, 190 ] gap> List( RootMod( 64, 1009 ) * RootsUnityMod( 1009 ), > x -> x mod 1009 ); [ 1001, 8 ] # set of all square roots of 64 mod 1009
gap3-jm